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# Factoring Trinomials

Sometimes we must use several methods to completely factor a polynomial.

Procedure â€” General Strategy for Factoring a Polynomial

Step 1 Factor out the GCF of the terms of the polynomial.

Step 2 Count the number of terms and look for factoring patterns.

Two terms: Look for the difference of two squares, the sum of two cubes, or the difference of two cubes.

Three terms:

â€¢ Try to factor using the patterns for a perfect square trinomial.

â€¢ If the trinomial has the form x2 + bx + c, try to find two integers whose product is c and whose sum is b.

â€¢ If the trinomial has the form ax2 + bx + c, try to find two integers whose product is ac and whose sum is b.

Four terms: Try factoring by grouping.

Step 3 Factor completely.

To check the factorization, multiply the factors.

Example 1

Factor: 108wx2 - 36wxy + 3wy2

Solution

 Step 1 Factor out the GCF.Factor each term. The GCF is 3w. Factor out 3w. 108wx2= 2 Â· 2 Â· 3 Â· 3 Â· 3 Â· w Â· x Â· x = 2 Â· 2 Â· 3 Â· 3 Â· 3 Â· w Â· x Â· x = 3w(36x2 - 12xy + y2) - 36wxy- 2 Â· 2 Â· 3 Â· 3 Â· w Â· x Â· y - 2 Â· 2 Â· 3 Â· 3 Â· w Â· x Â· y + 3wy2+ 3 Â· w Â· y Â· y + 3 Â· w Â· y Â· y Step 2 Count the number of terms and look for factoring patterns. There are three terms in 36x2 - 12xy + y2. The first term, 36x2, is a perfect square, (6x)2. The third term, y2, is a perfect square, (y)2. The middle term, 12xy, is twice the product of the squared terms: 12xy = 2(6x)(y) This matches the pattern for a perfect square trinomial. Substitute 6x for a and y for b: a2  2ab  b2 `(a  b)2 36x2 - 12xy + y2 = (6x)2 - 2(6x)(y) + (y)2 = (6x - y)2 Step 3 Factor completely. (6x - y)2 cannot be factored further.

Thus, the factorization is 3w(6x - y)2.

You can multiply to check the factorization.

Note:

Donâ€™t forget the GCF, 3w, that we factored out in Step 1.

Some polynomials cannot be factored using integers.

Example 2

Factor: x2 + 3x + 5

Solution

Step 1 Factor out the GCF.

There are no factors common to all three terms, other than 1 or 1.

Step 2 Count the number of terms and look for factoring patterns.

There are three terms in x2 + 3x + 5.

This trinomial has the form ax2 + bx + c.

To factor this trinomial we must find two integers whose product is 5 and whose sum is 3. Here are the possibilities:

 Product1 Â· 5 -1 Â· -5 Sum6 -6

There are no integers whose product is 1 and whose sum is 3.

So, the trinomial x2 + 3x + 5 cannot be factored using integers.

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